Here, \(f\)  is in \(B_r^{1+\beta }\)  for small \(r\) . We have \(\Phi (\mathbb {T})=\partial D_0\)  and we mention that the link between the regularity of the mapping and of the boundary is given by Kellogg-Warschawski's Theorem [1]}, [2]}. For \(w\in \mathbb {T},\)  a tangent vector to the boundary \(\partial D_0\)  at the point \(z=\Phi (w)\)  is given by
\(\overline{z^{\prime }}=-{\rm i} \overline{w}\overline{\Phi ^{\prime }(w)}.\) 
